#### Loops in AdS from conformal field theory

Received: May
Loops in AdS from conformal eld theory
Ofer Aharony 0 1 3 6 7
Luis F. Alday 0 1 3 4 7
Agnese Bissi 0 1 3 5 7
Eric Perlmutter 0 1 2 3 7
Andrew Wiles Building 0 1 3 7
Radcli e Observatory Quarter 0 1 3 7
0 Harvard University , Cambridge, MA 02138 U.S.A
1 Woodstock Road , Oxford, OX2 6GG , U.K
2 Department of Physics, Princeton University
3 Rehovot 7610001 , Israel
4 Mathematical Institute, University of Oxford
5 Center for the Fundamental Laws of Nature
6 Department of Particle Physics and Astrophysics, Weizmann Institute of Science
7 Jadwin Hall , Princeton, NJ 08544 U.S.A
We propose and demonstrate a new use for conformal eld theory (CFT) crossing equations in the context of AdS/CFT: the computation of loop amplitudes in AdS, dual to non-planar correlators in holographic CFTs. Loops in AdS are largely unexplored, mostly due to technical di culties in direct calculations. We revisit this problem, and the dual 1=N expansion of CFTs, in two independent ways. The rst is to show how to explicitly solve the crossing equations to the rst subleading order in 1=N 2, given a leading order solution. This is done as a systematic expansion in inverse powers of the spin, to all orders. These expansions can be resummed, leading to the CFT data for the spin. Our second approach involves Mellin space. We show how the polar part of the four-point, loop-level Mellin amplitudes can be fully reconstructed from the leading-order data. The anomalous dimensions computed with both methods agree. In the case of 4 theory in AdS, our crossing solution reproduces a previous computation of the one-loop bubble diagram. We can go further, deriving the four-point scalar triangle diagram in AdS, which had never been computed. In the process, we show how to analytically derive anomalous dimensions from Mellin amplitudes with an in nite series of poles, and discuss applications to more complicated cases such as the N = 4 super-Yang-Mills theory.
AdS-CFT Correspondence; Conformal Field Theory; Gauge-gravity corre-
1 Introduction
1.1
1.2
Setup
Summary of results
2
Crossing symmetry in the 1=N expansion
2.1
Setup
2.2 Implications from crossing at order 1=N 4
2.2.1
2.2.2
Truncated solutions at order 1=N 2
Solutions with in nite support at order 1=N 2
3
Loop amplitudes in AdS
Mellin amplitudes
UV divergences and freg
Examples
Large spin and the lightcone bootstrap
3.4
Enter crossing
4.1
The Casimir method
4
Solving the one-loop crossing equations
Basic idea
The basis h(n)(v)
Harmonic polylogarithms and the basis h^(n)( )
0(2;`) due to individual conformal blocks
3.1
3.2
3.3
4.2
4.3
5.1
Tree-level
One-loop
3.3.1
3.3.2
3.3.3
4.1.1
4.1.2
4.1.3
5.1.1
5.1.2
5.1.3
5.1.4
5.1.5
Summary
4 in AdS
5
Explicit examples
Expectations from the bulk
Solution from crossing
Comparison to AdS results
Relation to lightcone bootstrap
More general contact interactions
5.2
The four-point triangle diagram of AdS 3 + 4 theory
6
Computing anomalous dimensions from Mellin amplitudes
6.1
6.2
6.3
General problem
Application: 0(2;`) in 4
A remark on 3 theory
{ i {
40
43
44
7.1
7.2
Generalizations
Future directions
A.1 Degeneracies
B Explicit expansions
C
D
3 OPE data
A Operator content of the one-loop crossing equations
General expectations for UV divergences and the large n limit of
(2)
n;`
coupled theories of gravity in anti-de Sitter (AdS) space and conformal eld theories (CFTs)
with many degrees of freedom (\large N "). Perhaps the most fundamental element in
the holographic dictionary is that the AdS path integral with boundary sources is the
generating function of dual CFT correlation functions, thus making predictions for large
N , typically strongly coupled, dynamics. The 1=N expansion of the CFT correlators maps
to the perturbative expansion of AdS amplitudes, which is computed via the loop expansion
of Witten diagrams [1{3].
Such basics may seem hardly worth stressing: conceptually, the AdS side of this story
appears rather straightforward, and no di erent from
at space. However, perhaps
surprisingly, the physical content of the AdS perturbative expansion is poorly understood. Beyond
tree-level, the computation of AdS amplitudes is nearly unexplored, as almost nothing has
been computed. At one-loop and beyond, technical challenges inhibit brute force position
space computations: simple one-loop diagrams whose at space counterparts appear in
introductory quantum
eld theory courses, like the three-point scalar vertex correction and
the four-point scalar box diagram, have not been computed in AdS. Even at tree-level, the
original calculations [4{13] were impressive but arduous, and struggled to make manifest
the relation to CFT data; only recently have leaner, more transparent methods been
introduced, including Mellin space [14, 15] and geodesic Witten diagrams [16]. We emphasize
that these are not related to challenges of coupling to gravity: in an AdS e ective
eld
theory sans gravity, that can be dual to a decoupled sector of some CFT [17], the same
issues are present.
modern methods.
The purpose of this paper is to initiate a systematic exploration of loop amplitudes
in AdS, and of the dual 1=N expansion of holographic CFT correlation functions, using
There are (at least) two main reasons why one might be interested in this problem. The
tures: they relate loops to trees [18, 19], gravitational theories to gauge theories [20],
and have suggested a re-imagination of the role played by spacetime itself [21]. One is
led to ask: what is the organizing principle underlying the structure of AdS scattering
amplitudes? Given the existence of a well-de ned
at space limit of AdS (Mellin)
amplitudes [15, 22], the aforementioned structures should be encoded in, or extend to, the
analogous AdS amplitudes.
The second is to better understand the large N dynamics of holographic CFTs. The
marvelous universality of holographic large N CFTs is typically only studied at leading
order, dual to classical calculations in AdS. But the de nition of a holographic CFT must
hold at every order in the 1=N expansion. For instance, a large N CFT whose
entanglement entropy obeys the Ryu-Takayanagi formula [23], but not the
Faulkner-LewkowyczMaldacena correction term [24], cannot be dual to Einstein gravity coupled to matter. It is
the analogous correction that we would like to understand about the CFT operator product
expansion (OPE) data: namely, what the loop-level constraints are on operator dimensions
and OPE coe cients due to the existence of a weakly coupled gravity dual. In addition,
for given holographic CFTs whose planar correlation functions are known, we would like
to understand how to go to higher orders in the 1=N expansion.
While there is some work on one-loop AdS amplitudes [15, 25{27], some of which
we will make contact with later, loop physics in AdS has mostly been studied using other
simpler observables, speci cally the partition function (e.g. [28{33]). Interesting constraints
can indeed be extracted from the one-loop partition function | for example, in a
fourdimensional CFT, 1=N corrections to a and c can be computed by adding Kaluza-Klein
{ 2 {
contributions to the Casimir energy in global AdS5 | but correlation functions are much
richer objects. In particular, they depend on OPE coe cients and coordinates, and can
hence access Lorentzian regimes of CFT. Knowing loop amplitudes in a given bulk theory
would open the door to non-planar extensions of dynamical aspects of holography and the
conformal bootstrap [34{50].
While AdS loop amplitudes apparently pose di cult technical problems in position
space, there is reason for optimism. From the AdS point of view, given a classical e ective
action corresponding to the leading order in the 1=N expansion, one extracts the Feynman
rules, and computes loop diagrams accordingly. In this sense, loop amplitudes are xed
upon knowing all tree amplitudes, in principle. More precisely, the results of loop
computations are uniquely determined up to the need to x renormalization conditions for some
parameters; for any theory, renormalizable or not, only a
nite number of conditions is
required at any given loop order. The problem is to make the relation between loop-level
and tree-level AdS amplitudes precise, a la the Feynman tree theorem and generalized
unitarity methods for S-matrices.
How can quantitative progress be made? We will show that analytic solutions of the
conformal bootstrap for these four-point functions may be found at subleading orders in
the 1=N expansion. This may be viewed as either a CFT or a bulk calculation. The leading
order solutions for the connected four-point function of a single Z2-invariant scalar primary
were constructed in [51], where they showed that there is a one-to-one mapping between
those solutions and classical scalar eld theories on AdS space with local quartic
interactions. An important technical simpli cation of the leading order solutions is that they have
nite support in the spin, which makes manifest the analytic properties of the four-point
correlator. At subleading order this is no longer the case and the method of [51] does not
apply. Nevertheless, solutions can be constructed as a systematic expansion around large
spin, adapting the machinery of [52{54]. We nd that the solution to order 1=N 4 is fully
xed in terms of the data to order 1=N 2, to all orders in the inverse spin expansion.
Likewise we will show that the Mellin representation of the CFT four-point functions makes
it clear why and how higher orders in 1=N are determined by the leading-order result.
Furthermore, we will reconstruct the full one-loop Mellin amplitude for several examples.
1.1
Setup
Throughout the paper, we study an identical-scalar four-point function, hOOOOi, for a
scalar primary O of dimension
. This is determined in terms of an \amplitude" G(u; v)
of the two conformal cross-ratios, u and v (more details will be given in the next section).
G(u; v) admits an expansion in 1=N :
(0) is determined by mean eld theory, while G
(1)
G
(2)
G1 loop are the bulk tree-level and one-loop amplitudes, respectively.1 At every
Gtree and
1Following [51], we use the large N gauge theory notation 1=N 2 to stand for the small coupling in the
bulk. In a generic, full- edged holographic CFT, this stands for 1=c (even when c does not scale as N 2).
{ 3 {
(1.2)
(1.3)
order in the 1=N expansion, the amplitude is subject to the crossing equation
v G
(i)(u; v) = u G
(i)(v; u) :
We will also work with the Mellin representation of this amplitude, M (s; t), which admits
an analogous expansion.
Any large N CFT containing O also necessarily contains a tower of \multi-trace"
primary operators that are composites of O. The most familiar of these are the
doubletrace operators [OO]n;`, one for each pair (n; `) [15], whose de nition we recall below.
These acquire corrections to their conformal dimensions
n;` and squared OPE coe cients
OO[OO]n;` , at every order in the 1=N expansion:
to the one-loop Mellin amplitude M1[OOlo]op.
Mean eld theory determines a(n0;`), and the tree-level crossing equation determines n(1;`) and
a(n1;`) [51]. To solve the one-loop crossing equation is to derive n(2;`) and a(n2;`), in addition to the
OPE data for any other operators appearing at that order. We call the [OO] contributions
1.2
Summary of results
dimensions n(1;`)
In [51], the authors considered generalized free eld sectors of holographic CFTs in which
the only operators appearing at O(1=N 2) are the [OO] double-trace operators. Such setups
are dual to the simplest e ective
eld theories in AdS, namely,
4-type theories with no
cubic couplings. We will sometimes call these theories \truncated" theories on account
of the spin truncation
2p derivatives at the vertices has L = 2b p2 c.
(1)
n;`>L = 0 for some
nite L, as used in [51]; a bulk theory with
We note that in a truncated theory, the
double-traces are the only contributions to the full M1 loop: even at O(1=N 4), the O
OPE contains no single-trace operators by design, and no higher multi-trace operators by
O
necessity (a fact which we explain in appendix A).
One advantage of Mellin space is that it allows us to show explicitly how M1[OOlo]op can
in principle be derived directly from 1=N considerations alone. We show how large N
xes
the poles and residues of M1[OOlo]op, for any theory, in terms of the tree-level anomalous
. The location of the poles has been understood, and the residues derived
in a speci c example, in [15, 26, 27]; we show how to obtain this in general. We derive
the leading residue explicitly for a general theory (see (3.26)). In a truncated theory, the
leading residue is su cient to determine the large spin asymptotics of
passes a check against the lightcone bootstrap [35, 36] as applied to 4 theory.
(2)
n;` . The latter
In a general bulk theory such as
4, this stands for the four-point couplings (such as ), while three-point
bulk couplings scale as 1=N . We will freely interchange the labels tree-level/ rst order/O(1=N 2) throughout
the paper, and likewise for one-loop/second order/O(1=N 4).
{ 4 {
However, the above approach is somewhat clunky to implement and is not maximally
physically transparent. A more elegant, and more practical, approach is to solve the
oneloop crossing equations for
n(2;`) and an;`. This is tantamount to knowing the dual AdS
(2)
one-loop amplitude. The statement of bulk reconstruction is not just philosophical: we
can actually reconstruct M1 loop from OPE data, because they are related by a linear
Mellin integral transform. This is our proposed use for crossing symmetry: given leading
order OPE data, we solve the crossing equations at the next order, thus reconstructing
M1 loop for the dual AdS theory.
Let us now discuss what is involved in actually solving the loop-level crossing equations.
At one-loop, the tree-level data acts as a source in the crossing equation for G(2)(u; v), which
has a unique inhomogeneous solution. The freedom to add a homogeneous solution matches
expectations from the bulk, where one is free to modify the local quartic couplings at every
loop order: from [51], the correspondence between local quartic vertices and homogeneous
solutions to crossing follows. This pattern continues at higher orders.
In this work, we will focus on the anomalous dimensions
(2)
n;` . To actually compute
these from crossing, our main observation may be sketched as follows. In the regime
u
v
1, G(2)(u; v) contains terms of the form
xed by lower-order data, and is quadratic in the rst-order anomalous
dimensions n(1;`) (hence the log2(u)). By crossing symmetry, we also have
contribution to G
the precise equation is
At this stage we specify to truncated theories, where fan;`;
n;` g vanish above some nite
`. It is easy to see from the small u expansion that the term in (1.5) must come from a
(1)
(1)
(2)(u; v) that is linear in
(2)
n;` . For n = 0, where the analysis is simplest,
(1.4)
(1.5)
(1.6)
X a(00;`) 0(2;`)g2coll
+`;`(v) = 2f (v) log2(v) +
`
where gcoll
2
+`;`(v) is the lightcone, or collinear, conformal block, and \
" denotes
logarithmically divergent or regular terms. This is the desired equation for 0(2;`) in terms of
rst-order data 0;` .
(1)
nds 0;`
(2)
The solution of (1.6) is performed order-by-order in the large spin expansion: because
each term on the left-hand side diverges like log v, it must be that
and its large spin behavior is determined by matching to f (v). At leading order, one
0(2;`) 6= 0 for all `,
` 2 . A systematic expansion requires further development of the Casimir
methods utilized in [52{54], adapted now to this particular one-loop equation. Given a
large spin expansion of 0(2;`), a resummation down to
nite spin is possible when
Altogether, both the large and nite spin data constitute a holographic construction of the
one-loop amplitudes in the dual AdS theory that classically gives rise to the 0(1;`) used in
the crossing problem.
{ 5 {
In the above large spin analysis, we encounter an exciting mathematical surprise:
a certain class of harmonic polylogarithms forms a basis of solutions. In particular, if
we expand
(` +
)(` +
weight w
0(2;`) to nth order in inverse powers of the collinear Casimir eigenvalue J 2 =
1), then for integer
> 1, f (v) can be written as a linear combination of
2 + n harmonic polylogs, de ned in (4.16){(4.19).2 Harmonic polylogs are
speci ed by a weight vector, and only a speci c subclass of such functions appears in our
problem, namely those speci ed by the alternating w-vector ~w = (: : : ; 0; 1; 0; 1). Given
that multiple polylogs are ubiquitous in one-loop amplitudes in
at space, it is intriguing
to see some of them appearing in the construction of one-loop amplitudes in AdS via the
crossing equations.
Before showing our results for speci c theories, we should address an obvious question:
what happens when an AdS theory has a UV divergence? In particular, how is this visible in
the solutions to crossing? This has a satisfying answer. We expect to be able to cancel UV
divergences by adding a
nite number of local counterterms to our AdS e ective action
at a given loop order, just as in
at space. As explained in [51], local quartic vertices
with 2p derivatives generate anomalous dimensions only for double-trace operators of spin
`
2b p2 c. Therefore, on account of bulk locality of the divergences, we have a precise
prediction: when we compute a divergent one-loop bulk diagram via crossing, n(2;`) should
diverge for the above range of spins, where p is the number of derivatives in the counterterm.
Moreover, for any regularization, the divergence should be proportional to n(1;`). Analogous
statements apply at any loop order.
We demonstrate all of the above explicitly in the following two examples:
1)
4 in AdS.
The only non-trivial one-loop diagram is the bubble diagram of gure 1.
This is the one case where M1 loop is actually known directly from a bulk calculation,
performed in Mellin space in [26]: the authors used an AdS analog of the Kallen-Lehmann
representation to write the loop as an in nite sum of trees. Using our large spin data, we
reconstruct this amplitude in AdS3 and AdS5 for a
= 2 scalar, exactly matching the
result of [26]. (See (5.10){(5.13).)
Moreover, we show how to analytically compute 0(2;`) at some nite spins directly from
M1 loop itself. The results match the resummation of the large spin solutions to crossing.
This extraction had not been done previously | indeed, we know of no case in the literature
where OPE data has been analytically derived from a Mellin amplitude with an in nite
series of poles. We expect our regularization techniques to be useful more widely in the
world of Mellin amplitudes. The results at low spin align precisely with our UV divergence
expectations. The AdS3 theory is
nite, but the AdS5 theory requires a
Accordingly, 0(2;0) diverges in d = 4 (AdS5) but not in d = 2 (AdS3), and 0(2;`) for ` = 2; 4 is
4 counterterm.
nite in both cases. (See (5.3){(5.8).)
2) Triangle diagram of 3!
3 3 + 4!
4 4 in AdS.
This diagram, shown in
gure 1,
has never been computed, in any bulk spacetime dimension, as the trick of [26] does not
2The basis for
2= Z can be thought of as comprised of analytic continuations of harmonic polylogs to
non-integer weight. It would be interesting to formalize this.
{ 6 {
HJEP07(21)36
work here. Taking
= 2 for concreteness, we compute large and nite spin anomalous
dimensions from crossing (see (5.32){(5.35)), and reconstruct M1 loop. In d = 4, and in
the t-channel, say,
4 scalar, and is the rst computation of any such triangle diagram in any dimension.
We also give the analogous result in AdS3 in (5.41). It would be very interesting to discover
new tools for a direct evaluation in the bulk.
Overall, our work takes a step toward the nite N , Planckian regime by illuminating
the structure of the perturbative amplitude expansion in AdS and in large N CFT. As we
discuss in section 7, we believe that there is potential for the large N bootstrap to address
interesting questions beyond the realm of holography.
The paper is organized as follows. In section 2, we set up the crossing problem and
identify the key one-loop constraint. In section 3, we review the basics of Mellin amplitudes,
and use large N alone to explain how M1 loop is constrained by tree-level data, and to
construct the leading residue explicitly. In section 4, we develop the necessary tools for
solving the one-loop crossing equations in general. In section 5, we apply our machinery
to compute the bubble diagram of 4 in AdS, and the triangle diagram of 3 + 4 in AdS,
4
via crossing. In section 6, we explain quite generally how to compute low-spin anomalous
dimensions from Mellin amplitudes with an in nite series of poles; as an example, we apply
this to the one-loop bubble diagram in
. We conclude in section 7 with a discussion of
generalizations, applications to full- edged CFTs like N
= 4 super-Yang-Mills and the
d = 6 (2,0) theory, and other future directions. Some appendices include further details.
2
2.1
Setup
Crossing symmetry in the 1=N expansion
Consider a generic CFT with a large N expansion and a large mass gap. More precisely,
we assume there exists a \single-trace" scalar operator O of dimension
, and that all
other single trace operators acquire a very large dimension as N becomes large. This is
equivalent to considering a weakly coupled theory of a single scalar
eld in AdS, with
three-point couplings proportional to 1=N and four-point couplings proportional to 1=N 2.
Consider the four-point function of identical operators O. Conformal symmetry implies
hO(x1)O(x2)O(x3)O(x4)i = G(u; v)
2 ;
xj and we have introduced the cross-ratios u
Crossing symmetry implies
x213x224
x212x234 and v
(2.1)
xx212143xx222234 .
(2.2)
v G(u; v) = u G(v; u):
{ 7 {
We would like to study solutions to the crossing equation in a large N expansion, up to
O(1=N 4). As discussed in appendix A, up to this order and to inverse powers of the mass
gap, the operators appearing in the OPE of O with itself in a generic CFT include
O
O
1 + O + T
+ [OO]n;` + [T T ]n;` + [OT ]n;` ;
(2.3)
where 1 denotes the identity operator, T the stress tensor, and the double-trace operators
[OO]n;` are conformal primaries of the schematic form [OO]n;` = O
The presence of O and [OT ]n;` is forbidden in a theory with Z2 symmetry. Furthermore,
in the simplest setting we can ignore the presence of the operators including the stress
tensor; this is a good approximation when the self-couplings of the scalar are much larger
than its gravitational couplings, which is true in particular for a non-gravitational theory
on AdS. On the other hand, the presence of double-trace operators [OO]n;` is necessary
for consistency with crossing symmetry. Note that higher-trace operators will appear at
.
higher orders in the 1=N expansion, but not at O(1=N 4).
Let us for the moment focus on the simplest setting, in which the operators in the OPE
include only the identity operator and double trace operators [OO]n;`. This is relevant for
computing correlators of a
4 theory in AdS. In this case the four-point function admits
the following conformal partial waves decomposition:
G(u; v) = 1 + X
1
1
X
n=0 ` even
an;`u
n;` `
2
g n;`;`(u; v) ;
in which only even values of ` appear,3 and an;` denote the OPE coe cients squared
of [OO]n;` in the O
O OPE. The normalization of O has been chosen such that the
contribution of the identity operator is exactly 1. The conformal block for exchange of a
dimension
p, spin-` primary is written as
G p;`(u; v) = u 2 g p;`(u; v)
p `
so as to make manifest the leading behaviour for small u. Although most of the methods
of this paper will be general, we will mostly focus on d = 2 and d = 4 for de niteness. For
these cases the conformal blocks are given by (2.5) with
g p;`(z; z) =
g p;`(z; z) =
z`F p+` (z)F p ` (z) + z`F p ` (z)F p+` (z)
2
2
2
1 + `;0
z`+1F p+` (z)F p ` 2 (z)
z`+1F p ` 2 (z)F p+` (z)
2
2
2
2
;
where we have introduced the parametrization u = zz; v = (1 z)(1 z) for the cross-ratios,
and F (z)
2F1( ; ; 2 ; z).
3We assume that the identical external operators are uncharged under any global symmetries. Henceforth
we leave the even spin restriction implicit, and use P
1
P
At zeroth order in a 1=N expansion the four-point correlator (2.1) is simply the sum
over the disconnected contribution in all three channels:
This is consistent with the expected spectrum for double-trace operators at zeroth order
G
(0)(u; v) = 1 + u
+
u
v
:
(n0;`) = 2
+ 2n + ` ;
and leads to the following OPE coe cients [51]
a(n0;`) = 2Cn Cn+`;
a(n0;`) =
2(` + 1)(2
+ 2n + `
2)
(
1)2
Cn
1Cn+`1+1;
where we have introduced
Cn =
2
(
+ n) (2
+ n
1)
(n + 1) 2( ) (2
+ 2n
1)
:
We will study corrections to the four point function in an expansion in powers of 1=N
The dimensions and OPE coe cients of double-trace operators will have a similar expansion
n;` =
(n0;`) +
Let us start by recalling the analysis at O(1=N 2). Plugging the expansions for the
dimensions and OPE coe cients into the conformal partial wave (CPW) decomposition (2.4)
we obtain
G
(1)(u; v) =
(2.16)
Due to the convergence properties of the OPE, the right-hand side displays explicitly the
behaviour around u = 0. On the other hand, to understand the behaviour around v = 0 is
more subtle. Each conformal block behaves as
g p;`(u; v) v!0
a~ p;`(u; v) + ~b p;`(u; v) log v ;
(2.17)
where a~ p;`(u; v) and ~b p;`(u; v) admit a series expansion around u; v = 0. Hence each
conformal block diverges logarithmically as v ! 0. However, in nite sums over the spin
may generically change this behaviour. This will be important for us below.
In [51] a basis of solutions f n;`
(1)
; a(n1;`)g to the crossing equation (2.2) was constructed.
Each of these solutions has support only for a bounded range of the spin `. In this case,
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
(2.15)
HJEP07(21)36
the analytic structure around both u = 0 and v = 0 is manifest, and the crossing equation
v G
(1)(u; v) = u G
(1)(v; u) can be split into di erent pieces, proportional to log u log v,
log u, log v and 1 (times integer powers of u and v). In [51] it was argued that there is a
one-to-one map between this basis of solutions to crossing and local four-point vertices in
a bulk theory in AdSd+1. Furthermore, let us mention that in Mellin space these solutions
correspond simply to polynomials with appropriate symmetry properties. The degree of
the polynomial determines for which range of spins the corrections f n;`
from zero. Let us stress that these \truncated" solutions are consistent with crossing only
(1)
; a(n1;`)g are di erent
in the minimal set-up, in which only the identity and double-trace operators are present in
the OPE O
O. Later we will discuss what happens in more general cases.
The aim of the present paper is to extend those solutions to consistent solutions to
(1)
crossing at order 1=N 4. We will assume the leading order solutions f n;`
(1)
; an;`g as given,
(2)
; a(n2;`)g. Plugging the expansions (2.14) into the
and analyze consistency conditions on f n;`
CPW decomposition (2.4) we obtain
8 n;`
12 a(n1;`) n(1;`) log u +
( n(1;`))2 log2(u) + 2 log u
g2 +2n+`;`(u; v) :
(2.18)
(2.19)
; a(n1;`)g. The contribution from the other lines is uniquely
solution at order 1=N 2 and can be viewed as a source, or an inhomogeneous term, for the
xed in terms of the
crossing equation
v G
(2)(u; v) = u G
(2)(v; u) ;
interpreted as an equation for f n;`
(2)
; a(n2;`)g. The analysis of this equation is much harder than
the analysis at order 1=N 2, since, as we will see momentarily, consistency with crossing
implies that f n;`
(2)
; a(n2;`)g are di erent from zero for arbitrarily large spin. We will focus here on
certain unambiguous contributions to the source terms, and understand their implications
for the solution to the crossing equation.
2.2
Implications from crossing at order 1=N 4
Let us focus on a speci c contribution to G(2)(u; v), which is the coe cient of log2(u):
G
This contribution is unambiguously xed in terms of the leading order solution. We can
already make the following simple observation. Under crossing symmetry this term will
map to a term with a divergence log2(v) as v ! 0. Since each conformal block diverges at
most logarithmically in this limit, such a contribution must come from an in nite sum over
the spin, for a given twist. Hence it follows that the solution f n;`
; an;`g must be di erent
from zero for arbitrarily large spins, even if the solution at order 1=N 2 is truncated. From
now on, it is convenient to restrict our considerations to truncated solutions at order 1=N 2.
(2)
(2)
More general solutions will be studied in section 2.2.2.
If the solution at order 1=N 2 truncates at spin L, we have:
G
where we have used the fact that the sum over spins truncates. f (u; v) and g(u; v) admit
a series expansion in u; v with integer powers, and can be computed in terms of the given
leading order solution. As a consequence of crossing symmetry, G
contain the following terms:
(2)(u; v) should also
G
(2)(u; v) = u log2(v) (f (v; u) log u + g(v; u)) +
(2.22)
where the dots denote contributions proportional to log v, or analytic at v = 0. Given that
the support of f n;`
(1)
; a(n1;`)g involves a
cannot generate a log2(v) behaviour, since each conformal block diverges at most
logarithnite range of the spin, the last two lines of (2.18)
mically. Hence
(2.21)
n 12 a(n0;`) n(2;`)g2 +2n+`;`(u; v)
log2(v)
= f (v; u) ;
(2.23)
and there is a similar equation involving the OPE coe cients a(n2;`). (2.23) should be
interpreted as an equation for n(2;`), with the right-hand side f (v; u) completely xed in terms of
the solution at order 1=N 2. As already mentioned, since we need to reproduce an enhanced
divergence on the left-hand side, we need to sum over an in nite number of spins.
Furthermore, the divergence will arise from the region of large spin. In section 4 we will adapt the
algebraic method developed in [53, 54] to determine the necessary large spin behaviour on
n(2;`) in order for (2.23) to be satis ed. The nal answer is an expansion of the form
n(2;`) =
c(n0)
`2
1 +
b(n1)
`
+
b(n2)
`2 +
!
;
(2.24)
where all the coe cients of the expansion are actually computable. Hence we conclude
that (2.23) actually
xes
n(2;`) up to solutions which decay faster than any power of the
spin. Notice in particular that, from this point of view, we cannot expect to do any better,
since there is always the freedom to add to any solution of (2.19) a truncated solution
which solves the homogeneous crossing equation (the same equation appearing at order
1=N 2). In section 5 we will study several examples. For these examples we will actually be
able to do much more: we will be able to re-sum the whole series (2.24), and extrapolate
the results to nite spin.
log u + aT
v
u
d 2 log u +
2
;
which lead [35, 36, 55] to the following large spin behaviour for the anomalous dimensions
of double-trace operators:
(1)
n;`
(1 +
) +
`adT2 (1 +
) :
This implies, in particular, that the leading order solution has in nite support in the spin.
Single-trace operators can be seen as sources for the crossing equations, which are otherwise
homogeneous. The general structure of the solution at order 1=N 2 is then the sum of a
solution to the equation with sources, with the behaviour (2.27), plus any of the truncated
solutions studied above. Although any full- edged conformal eld theory contains the
stress tensor, we will discuss its inclusion in a separate publication. In this paper, instead,
we will consider only the presence of O. This is relevant for correlators of 3 theory on
AdS. In this case
where n(1;`); 3
n(1;`) = a
(1);trunc is any one of the truncated solutions.
As before, we can compute the piece proportional to log2(u) at order 1=N 4. We obtain
G
Let us now discuss the more general situation, in which the OPE of O with itself also
includes single-trace operators. The two most important examples are O itself, of dimension
, and the stress tensor T , a spin two operator of dimension
T = d and twist
T
` =
d
2. These single-trace operators enter in the OPE decomposition with OPE coe cients
squared of order 1=N 2. In these cases G
(1)(u; v) contains the following terms
G
(1)(u; v) = a u 2 g ;0(u; v) + aT u d 2 2 gd;2(u; v) +
Under crossing symmetry these map into terms of the form
where now the sum over ` is not truncated. As we have already discussed, for a truncated
solution the small v behaviour is simply proportional to log v, as for a single conformal
block. In the case at hand, however, since the sum over the spin now does not truncate,
we get an enhanced behaviour. More precisely, (2.27) leads to
G
Under crossing symmetry this contribution maps to itself, so that h(u; v) = h(v; u). This
is a consequence of crossing and the OPE expansion, and is completely independent of the
new data f n;`
(2)
(2)
; an;`g at order 1=N 4. In addition, as in (2.21), the sum above will contain
contributions proportional to log v
However their computation is more subtle than before: one needs to perform the sum over
the spin, and then expand for small v. Both the truncated and non-truncated parts of the
solution will contribute to this term. The analysis of the crossing equations is now more
complicated. Under crossing the term (2.31) maps to a term proportional to log u log2(v).
However, as the support of the solution at order 1=N 2 is in nite, several terms in (2.18)
can produce an enhancement log2(v), and not only those involving n(2;`). While this general
case can also be analysed, note that the contributions from crossed terms to ( n(1;`))2, of the
form (2a
(1);trunc) are much simpler to analyse. These crossed terms have a nite
support, and their contribution to
n(2;`) can be computed exactly as explained above. We
will discuss the interpretation of these contributions, and will compute them for speci c
examples, in section 5.
3
Loop amplitudes in AdS
The subleading solutions discussed in the previous section may be interpreted as one-loop
contributions to correlation functions in AdS. We now turn to constraining the general
form of loop-level AdS amplitudes by studying features of the large N expansion. We
will employ the Mellin representation. One of the advantages of Mellin space is that AdS
amplitudes have a transparent analytic structure as a function of the Mellin variables.
This has been utilized in [15] to write down compact and intuitive forms for tree-level
Witten diagrams, and we will do the same here at one-loop. See [14, 15, 26, 27, 56{62] for
foundational work, and [22, 62{68] for some recent applications, of Mellin space in CFT.
3.1
Mellin amplitudes
spondence.
We now give a crash course in Mellin amplitudes in the context of the AdS/CFT
corre
Consider the four-point function of identical operators hO(x1)O(x2)O(x3)O(x4)i,
related to an amplitude G(u; v) by (2.1). By a double Mellin transform, we can trade G(u; v)
for the Mellin amplitude, M (s; t), de ned to be
t. The two integration contours run parallel to the imaginary axis,
such that all poles of the gamma functions are on one side or the other of the contour.4
The product of gamma functions is totally symmetric in permutations of (s; t; u^). Crossing
symmetry of G(u; v) then implies total permutation symmetry of M (s; t; u^):
M (s; t) = M (s; u^) = M (t; s) :
(3.2)
4The following formulae are specialized to the case of identical external operators, although many also
hold for pairwise identical operators. A summary of the relevant formulae can be found in appendix A
of [59]. Their conventions are the same as in (3.1) up to a shift shere = sthere + 2 .
gcopll;`(v) is the collinear block,
The Mellin representation for all g(mp);`(v) is
g(mp);`(v) =
Second, Q`;0(s; p) takes the explicit form
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
For pairwise identical external operators as here, the Q`;n(s; p) do not depend on the
external dimensions.
We will make use of the following facts about the Q`;n(s; p). First, they are intimately
related to the Mellin transform of the conformal blocks for exchange of a twist- p operator.
In the lightcone expansion u 1, the blocks take the form (2.5) with
g p;`(u; v) =
1
m=0
X umg(mp);`(v) :
gcopll;`(v) = (1
v)` 2F1
p + `
2
;
p + `
2
; p + `; 1
v
:
In a CFT with a weakly coupled AdS dual, the conformal block decomposition of
G(u; v) translates into a sum of poles in M (s; t). In a given channel, say the t-channel,
the amplitude M (s; t) has poles in t at the twists of exchanged operators, and the residues
encode the OPE coe cients:
where the exchanged primary operator Op has twist p =
p
`p. The pole at t = p + 2n
captures contributions of the twist-( p + 2n) descendants of Op: schematically, these are
The residues Q`;n(s; p) are the Mack polynomials, whose precise de nition can be found in
appendix A of [59]. They have a spin index ` and a \level" n, and they depend on both the
external and internal operator data. We will nd it convenient to work with a \reduced"
polynomial, Q`;n(s; ), related to Q`;n(s; ) in general by [59]
Q`;n(s; p) = Q`;n(s; p)
2 ( p + `)( p
1)`
v (s+ p 2 )=2Q`;m(s
p; p) 2
2
s
2
2 s
2
:
Q`;0(s; p) =
2
` p 2
2 `
( p + `
1)` 3F2
`; p + `
1;
s
2
; p
2
2
; p ; 1 :
This has the useful property that
Q`;0(s
p; p) = ( 1)`Q`;0( s; p) ; ` 2 Z :
These obey an orthogonality relation [59], which can be written5
Given some amplitude expanded in the lightcone regime of small u and xed v, this relation
allows one to strip o the coe cient of the leading-twist, spin-` lightcone block g2coll
+`;`(v)
due to the exchange of [OO]0;`.
We now develop the AdS loop expansion of the connected piece of G(u; v) and M (s; t),
corresponding to the 1=N expansion of some holographic CFT:
where
An(v) =
of our knowledge.
1
4 in!2
Z i1
i1
1
1
1
1
To set the stage for M1 loop, we need to review the structure of Mtree.
We are interested in paradigmatic large N holographic CFTs which have a large gap in
their spectra, or generalized free eld sectors thereof. These are dual to weakly coupled
gravity, coupled to a
nite number of light
elds. The spectra of these theories consist
of \single-trace" operators Oi and their \multi-trace" composites [OiOj ]; [OiOj Ok], etc.,
that are dual to single-particle and multi-particle states in the bulk, respectively.
As
discussed above, the CFT conformal block decomposition of Gtree only includes single-trace
and double-trace exchanges.
There are two salient points about Mtree. The rst is that its only poles come from the
single-trace exchanges of Gtree. These each contribute as in (3.3). The second is that the
double-trace exchanges of Gtree are accounted for by the explicit 2 factors in the Mellin
integrand (3.1), one for each channel, which have double poles at
= 2
+ 2n. This makes
explicit a fact about holographic CFTs: at tree-level, the single-trace OPE data completely
determine the double-trace OPE data, up to the presence of regular terms in Mtree.
The gamma function residues include a log u term and a term regular at small u,
(3.10)
`;`0 :
(3.11)
(3.12)
(3.13)
(3.14)
:
(3.15)
t=2 +2n Gtree(u; v) = u +n An(v) log u + Bn(v) ;
Res
ds v
(s+2n)
2
Mtree(s; 2
+ 2n) 2
s + 2n
2
2
2
2
s
5Analogous orthogonality relations exist at higher n but have not been calculated explicitly, to the best
The Mellin representation of g2coll
+`;`(v) may be written
where we have introduced a convenient combination for future use,
gcoll
2
+`;`(v) = ( 1)`16 d ;`
Upon using the orthogonality relation (3.11), one nds the explicit formula [62]
0(1;`) =
A similar analysis allows one to extract a0;` . For higher n, one must deconvolve the
subleading corrections g(m;`)(v) to the small u blocks, from the leading contributions coming
from n > 0 double-trace primaries. Expressions and an algorithm for computing g(m;`)(v)
can be found in [53].
3.3
fall into two categories:
We now turn to M1 loop. In a general CFT, this may receive various contributions. These
Bn may be extracted similarly. Matching this to (2.16), one can extract n(1;`) and a(n1;`) by
picking o the contribution proportional to the appropriate conformal block in the u
expansion. The An log u terms contain n(1;`), and the Bn terms contain a(n1;`). The extraction
of the leading-twist double-trace operator data, like 0(1;`), is especially simple: from (2.18),
1
we require
First, there are loop corrections to tree-level data. This includes mass, vertex and wave
function renormalization of elds already appearing at tree-level; that is, O(1=N 4) changes
to the norms, dimensions and OPE coe cients of CFT operators appearing in the planar
correlator. Corrections to the OPE data of single-trace operators can arise, but they can
be easily taken into account by expanding the leading order solutions, and we will assume
for simplicity that they vanish. Note that in any case these cannot be determined by the
crossing equations, which have solutions for any such data.
Second, as discussed in appendix A, there are new operator exchanges that do not
appear at tree-level, due to large N factorization. A simple example in a theory of gravity
coupled to a scalar eld is the appearance of two-graviton intermediate states, dual to
[T T ]-type double-trace operators, in the scalar correlator hOOOOi.
A universal contribution in any holographic CFT is the next-order correction to the
tree-level [OO]n;` OPE data, namely, n(2;`) and a(n2;`). Let us write the double-trace piece of
the total one-loop amplitude as
M1[OOlo]op(s; t) :
We note that in simple AdS e ective theories like
4 dressed with any number of deriva
tives, this is the full amplitude. More precisely, for any theory in which no single-trace
operators appear in the OPE (dual to theories in AdS with no cubic vertices), and in which
there are no extra double-trace operators in the OPE (dual to the absence of four-point
couplings to other elds in AdS), we have
M1 loop(s; t) = M1[OOlo]op(s; t):
(3.21)
in section 7.
(1)
n;` and an;`.
When f ig 2 2Z in more general theories, there are similar simpli cations, as we discuss
We now establish the following simple but powerful claim: all poles and residues of
M1[OOlo]op are completely xed by tree-level data. It follows that n(2;`) and a(n2;`) are xed by
Recall that the contribution of [OO]n;` to G(2) takes the form given in (2.18):
G
The point is that there is a log2(u) term whose coe cient is completely xed by tree-level
data. In order to correctly produce this term at each power u +n (n = 0; 1; 2; : : :), two
things must happen:
1) M1 loop must acquire simple poles at
= 2
+ 2n for n = 0; 1; 2; : : :.
2) The residues are xed by n(1;`) so as to match (3.22).
This is true in each of the s; t; u^ channels, so we can focus on just one, and trivially add
the crossed channels to get the full M1[OOlo]op. Showing the t-channel for concreteness, we
have thus determined that
M1[OOlo]op(s; t) =
1
X
n=0 t
Rn(s)
(2
+ 2n)
+ freg(s; t) + (crossed)
(3.23)
for some residues Rn(s). This argument does not determine any possible regular terms in
M1[OOlo]op, so we have allowed for a function freg. We drop this for now, but will return to
it shortly; as we will see, freg is not unique.
To determine the residues Rn(s), we use the same technique as at tree-level. We have
t=2Re+s2n [G1 loop(u; v)] = u +n An(v) log2(u) + Bn(v) log u + Cn(v) ;
(3.24)
where An; Bn; Cn are easily determined by plugging M1 loop of (3.23) into the Mellin
amplitude formula (3.1). To
x the Rn(s) we insist upon equality of An with the log2(u)
term in (3.22). Given the Mellin representation (3.8) of the conformal blocks in the u
1
expansion, this xes the Rn(s) completely for every n.
For example, the leading residue R0(s) is determined by the following equation:
Using the Mellin representation (3.17) of g2coll
+`;`(v) determines R0(s) to be
(3.25)
(3.26)
HJEP07(21)36
1
`=0
R0(s) =
X a0;` ( 0(1;`))2d ;` Q`;0( s; 2 ) ;
(0)
where d ;` was de ned in (3.18), and Q`;0( s; 2 ) is the polynomial (3.9) at intermediate
twist 2 . Note that in the formula for R0(s), the coe cients of Q`;0( s; 2 ) are manifestly
positive. Higher Rn(s) can, with some work, be extracted similarly. By matching Bn
in (3.24) to the log u terms in (2.18), one can compute n(2;`), as we will show in an explicit
example shortly.
UV divergences and freg
We now return to the physics of the function freg in (3.23).
The rst point to note is that (3.23) is a solution to crossing for any
permutationsymmetric freg. The minimal solution is freg = 0. Indeed, freg re ects the freedom to add
a homogeneous solution to the second-order crossing equations (2.18). Such solutions sit in
one-to-one correspondence with quartic contact interactions in AdS; in Mellin space, these
are simply crossing-symmetric polynomial amplitudes [15, 51]. So we should think of freg
as a choice of one-loop renormalization conditions for the quartic part of the e ective action
for the light elds in AdS, dual to a choice of one of the in nite solutions to the one-loop
crossing equations that di er by polynomials, i.e. nite local counterterms in AdS.
What happens when the bulk theory is one-loop divergent? In this case, one must
include in the bulk some diverging local counterterms to restore
niteness. Due to their
locality, these again appear in the function freg. This was explained in general terms in the
Introduction; for more discussion, see appendix C. In the explicit results for scalar theories
that will follow in section 5, we will see very nicely in detail how bulk UV divergences show
up in the one-loop CFT correlators.
For all of these reasons, freg is not unique, and may sometimes not be nite before
renormalizing the bulk theory. We note that various high-energy limits, such as the Regge
limit of large s and xed t < 0, may place some constraints on freg, see e.g. [48].
3.3.2
Let us treat some simple and instructive examples.
The rst is
4 theory in AdS. There is a single non-trivial one-loop diagram, the
bubble diagram, in each channel. (There are also diagrams which lead to mass and wave
function renormalization of , but these only serve to renormalize Mtree, which is anyway
constant in this case.) On the CFT side, as explained earlier in section 2, there are only
This matches (5.13). This is a substantial check on the match between CFT and AdS: we
have successfully reconstructed the
4 one-loop amplitude from the conformal bootstrap.
The expected UV divergence structure is apparent in the above: at large m, one has
^
Rm
This leads to a divergence in the sum over m for d
3 | that is, in AdSD 4 | with a
logarithmic divergence at the critical dimension dc = 3. We note for later that the d = 2
amplitude can be resummed to yield the t-channel amplitude
n2`+1 ;
n2`+2 ;
for d = 2,
for d = 4.
It is not obvious that the low-spin 0(2;`) as computed from the Mellin amplitudes above
will match those from the crossing problem. For this reason, we would like to analytically
compute 0(2;`) for ` = 2; 4 directly from (5.14). This has never been done. Doing so requires
new techniques that should be useful more generally for extracting anomalous dimensions
from Mellin amplitudes with an in nite series of poles in a given channel.
We devote section 6 to this endeavor. The end result is a perfect match for ` = 2; 4 in both d = 2 and d = 4.
5.1.4
Relation to lightcone bootstrap
Note that in both d = 2 and d = 4, the anomalous dimensions are negative and
monotonically increasing with `:
0(2;`) < 0 ;
We have checked this behavior to higher ` as well. These properties must in fact hold
for all ` and all (unitary)
, as can be explained by resorting to Nachtmann's theorem
and the lightcone bootstrap. The basic point is that, because 0;`>0 = 0, these one-loop
anomalous dimensions are actually the leading corrections to the mean eld theory result.
(1)
The O
O OPE is re ection positive and contains only even spin operators, which implies
monotonicity via the arguments in [35, 36, 43]; moreover, the negativity follows from the
large spin asymptotics given in (3.40).
5.1.5
More general contact interactions
We could also consider more general solutions, where n(1;s) is di erent from zero also for s 6=
0. This corresponds to (@ )4-type theories, etc. Let us analyse the issue of divergences in
this case. For instance, for s = 2 all the explicit results we have obtained (too cumbersome
to be included here) are consistent with
(5.20)
On the other hand, n(1;s) has generally an enhanced behaviour, with respect to the
solution studied above. For instance, an irrelevant interaction such as (@ )4 leads to a
behaviour n;s
(1)
nd+1, see [60], as expected from the analysis of appendix C. For d = 2
this implies that the resulting 0(2;`) will be convergent only for ` > 2. For d = 4, the result
will be convergent only for ` > 4.
The four-point triangle diagram of AdS
We now consider the following (Euclidean) AdSd+1 e ective theory:
Lbulk =
1
2
m2 2 +
1
2
3!
On the crossing side, we now consider in more detail the solution
discussed in
section 2, together with a truncated
4 solution
with support only on operators
with ` = 0. We will compute the crossed term contribution, proportional to
This only receives spin-0 contributions, and computes the sum over channels of the
fourpoint triangle diagram in AdS5, as shown in gure 3.14
For de niteness, we rst take d = 4 and
= 2, hence m2 =
(
d) =
units. The rst-order data needed on the right-hand side is
=
2
2(2
7(1 + n)2)
3 (1 + n)(3 + 4n(2 + n))
;
n;0
;
where n(1;0); 3
is computed in appendix D. We note that n(1;0); 3 (1); 4
n;0
leads to a convergent contribution for (5.26) even for the case ` = 0, as expected from the
1 for large n. This
bulk. For the rst few spins we obtain
In addition, the full theory will have a term proportional to 24 from the bubble diagram of
gure 2, which is exactly as before, plus a term proportional to
43 from the box diagram
of gure 1, which is harder to compute.
14For general eld theories on AdS, in which each vertex comes with an independent coupling constant,
it is easy to identify the number of vertices of each type in the diagrams that contribute to each term in the
crossing equation and in the anomalous dimensions. In speci c theories like supergravity, many di erent
coupling constants are related and it is di cult to separate the contributions from individual diagrams; the
solution to crossing just gives the full 1-loop amplitude.
amplitude for a m2 =
AdS3 is given in (5.41).
hence m2 = 0 in AdS units. Now we use
4 scalar in AdS5 is given in (5.38). The amplitude for a massless scalar in
The same analysis of (5.26) can be done also for d = 2. We again take
= 2, and
n;0
=
2(4n + 5)
2
3 (n + 1)(n + 2)(2n + 3)
;
n;0
3
= 4 (2n + 3)
:
Analogously to the previous case, we compute the value of (2) for some value of the spin
4 case, we may use the 1=J expansion of 0(2;`) to reconstruct the bulk
amplitude. The 1=J expansion used to derive the above results for d = 4 is
1 +
18 1
Analogously to the 4 case, we nd the following result:
with
M1 loop(s; t) =
1
X
m=0 t
Rm
(4 + 2m)
+ (crossed) ;
Rm =
3(10 + 7m)p
(1 + m) 23 4 :
( 52 + m)
(5.31)
(5.32)
(5.33)
(5.34)
(5.36)
(5.37)
The amplitude can be resummed: in the t-channel, say,
as quoted in the introduction. This gives a prediction for the triangle Witten diagram
for a m2 =
4 scalar in AdS5. Note the striking similarity to the
4 bubble diagram for
d =
= 2 in (5.21), which is completely unobvious from the spacetime perspective.
We can perform the same analysis for d = 2 and
= 2. The only di erence with
respect to the previous case is n(1;0); 3 , which is computed in appendix D. The 1=J
expansion is
1 +
2 1
25 J 2
8 1
In this case, we nd (5.36) with
We can again resum the amplitude and obtain, in the t-channel,
6H(1
t
2
t
2
t
where H(x) denotes the harmonic number of argument x, de ned for x 2= Z via the relation
to the digamma function, H(x) =
(x + 1) + . This gives a prediction for the triangle
Witten diagram for a massless scalar in AdS3.
6
Computing anomalous dimensions from Mellin amplitudes
In this section, we develop techniques for analytically computing double-trace anomalous
dimensions from Mellin amplitudes. In particular, we focus on cases where the amplitude
has an in nite series of poles. This necessarily occurs at one-loop as explained in this work,
but also occurs in the tree-level exchange diagram of 3 for generic
, or the tree-level
exchange of a dimension
0 scalar between external dimension
scalars, where
0
2
2Z. To our knowledge, the only treatments that have appeared in previous literature
=
2
deal with
nite sums of poles. As an application, we derive the one-loop
4 anomalous
dimensions for ` = 2; 4, described in the previous section.
6.1
General problem
the form
Consider an exchange amplitude between identical external scalars of dimension
, of
for some residues Rm and some internal dimension
0. If this represents a tree-level
0 scalar primary, say, the residues are Rm(t) / Qm;0(t; ).
I`( ; )
2 s
2
2
2
2
s
3F2
The double-trace anomalous dimension 0;`>0 receives contributions from the two crossed
channels: from (3.19),
Rm(2 )I`( ; 0 + 2m) :
(Recall that the direct-channel amplitude only contributes to ` = 0, since it evaluates to a
constant on the pole at 2 .)
We split the analysis into two parts. First, we evaluate I`( ; ), i.e. we determine
the contribution to the anomalous dimension from a single pole. Next, we perform (6.3),
summing over contributions from all poles.
To evaluate I`( ; ), we close the contour to the left, picking up an in nite series of
poles at s = 0; 2; 4; : : :. The resulting in nite sums can be regularized using Hurwitz zeta
functions. Upon looking at several examples, one infers the following structure for
(6.2)
(6.3)
2 Z:
(6.4)
(6.5)
(6.6)
(6.7)
(6.8)
2
X
n=nmin
I`( ; ) =
Pn( )
2
nmin =
(` + 2
4) :
n;
2
=
Bn+1( 2 ) :
n + 1
where
The Pn( ) are degree-(n
nmin) polynomials in , and P1( ) = 0. All but the
2; 2 terms
reduce to Bernoulli polynomials in ,
Since Bn+1 is degree-(n + 1), we can rewrite the form of I` as
I`( ; ) = P`+2
3( ) + R`+2
2( )
2;
2
;
where Pm and Rm are polynomials of degree m.15 Note that (2; x) = 0(x) = d2x log (x).
Now we want to sum over all poles at
=
0 + 2m. Plugging (6.7) into (6.3),
(1)
15There is some potential ambiguity in these polynomials; this can be xed in a given case by comparing
to numerical integration.
two regularization methods.
P`+2
0 as xed. The second term is somewhat tricky. To proceed we employ
The rst is an exponential regularization. This is useful when evaluating the sum over
1
X
Rm(2 )P`+2
3(m)e
m :
0( 0 + m) =
0
dt
t e t( 0+m)
1
e t
:
Performing the sum and expanding near
= 0, the prescription is to keep the nite term,
dropping terms that are power law divergent.
The second is an integral regularization. This is useful when evaluating the sum over
2(m) 0( 0 + m). Speci cally, we turn to the integral representation of 0( 0 + m),
Swapping the order of the sum over m and the integral, performing the sum over m, and
then performing the integration analytically, the prescription is to keep the nite term,
dropping terms that are power law divergent near t = 0.
We have checked that these two methods agree in several examples in which both can
be carried to the end, e.g. a tree-level scalar exchange with
= 2; 0 = 3.
We now apply the above to compute 0(2;`) for ` = 2; 4. To make contact with the previous
section, we take d = 4;
= 2, and use the residues (5.11). From (6.3),
(2)
RmI`(2; 4 + 2m) :
(6.11)
First we compute I`(2; 4 + 2m). Let's rst focus on ` = 2. Closing the contour in (6.2)
to the left, we pick up the poles at s = 0; 2; 4; : : :, which yields the following in nite sum:
I2(2; 4 + 2m) =
k=0
X1 (k + 1) 15k3 + 5k2(4m + 13) + k(40m + 86) + 22m + 38
6(k + m + 2)2
We can regularize this using Hurwitz zeta functions of the form
(y; m + 2) for y =
2; 1; 0; 2. Comparing this result to numerical integration, we nd that an extra
additive polynomial is required. The end result is
where
I2(2; 4 + 2m) ! P(m) + R(m) 0(m + 2) ;
P(m)
R(m)
1
36
1
6
(30m3 + 75m2 + 71m + 23) ;
(m + 1)2(5m2 + 10m + 6) :
(6.9)
(6.10)
: (6.12)
(6.13)
(6.14)
We now need to perform the sum (6.11). We rst do the sum over P(m) using an
exponential regulator. The sum yields a linear combination of generalized hypergeometric
functions; upon expanding in small and keeping the nite term, we get
1
X
m=0
P(m)Rme
m
76
! 105
2
:
Next, we use the integral regularization on the sum over the R(m) term. After performing
the sum inside the integral, we have
1
X
m=0
R(m)Rm 0(m+2) =
0
t
1 e t
Performing the integral, and keeping the nite terms,
1
X
m=0
R(m)Rm 0(m + 2) !
583 + 174 2
3465
2
:
Adding this to (6.15) and multiplying by (-2) to obtain (6.11), the nal result is
This agrees with equation (5.7). An analogous procedure can be carried out for ` = 4. The analog of (6.13) is with
I4(2; 4 + 2m) ! P(m) + R(m) (2; m + 2) ;
P(m) =
R(m) =
1
60
630m5 + 2835m4 + 6195m3 + 7350m2 + 4579m + 1159
1800
Carrying out the sum over m using the above techniques, we get
1
X
m=0
P(m)Rm ! 214500
104267 2
1
X
m=0
R(m)Rm 0(m + 2) !
2 83636 + 18825 2
1126125
Adding the two numbers and multiplying by (-2), this agrees with (5.8). We have repeated
all of the above for d = 2, nding agreement there as well.
To summarize, the results of this subsection give further con rmation that our
solution to the crossing problem is equivalent to a direct computation of 4 one-loop Witten
diagrams in AdS. We reiterate that this agreement acts as a check on a match between
two independent techniques used to derive anomalous dimensions: on the one hand, the
large spin resummation technique used in the crossing problem, without reference to any
amplitude; and on the other, the techniques of this section used to extract low-spin data
from M1 loop.
(6.15)
(6.16)
(6.17)
(6.18)
(6.19)
(6.20)
2
:
(6.21)
A remark on
3 theory
There is a small subtlety when computing
leading pole of an exchange diagram in
a pole at twist
in all three channels is
n(1;`) for 0 =
. A common example is for the
3 theory. The fully symmetrized contribution of
s
1
If we evaluate the s and u^ poles on the 2 double-trace pole at t = 2 , they cancel. Thus,
naively, so do the s- and u^-channel contributions to
6
add, just as they do for
. To get around this, one can simply deform the internal
dimension by a small amount,
+ , perform the computation in which the two
example, see appendix D for the computation of n(1;`) in 3 theory for
= 2.
channels add, and then take
! 0. A spacetime computation con rms this result. For
(1)
n;` . However, they are supposed to
In this paper we initiated an analysis of large N CFT four-point correlators at
next-toleading order in 1=N , which map by the AdS/CFT correspondence to one-loop diagrams
in AdS space. We presented general methods to analyze correlation functions at this order,
and implemented them explicitly for two examples: a 4 theory in the bulk, and a triangle
diagram in a
3 + 4 theory in the bulk.
There are various levels of extension of what we have done here, most of which are
needed in order to study the 1=N expansion in more generic, full- edged holographic CFTs.
We rst discuss some of these, and then move on to broader future directions.
7.1
An immediate priority, and a necessary step toward solving bona de CFTs, is to
solve the crossing equations when the OPEs contain single-trace operators. In the
single-scalar theory, this would yield a computation of the scalar box diagram in AdS
3 theory.
We could also allow exchanges of operators with spin. These should present some
technical complications, but we do not expect them to lead to any qualitative changes.
A particularly important version of this is to incorporate the stress tensor, which
allows us to access graviton loops.16
One would also like to extend our methods to include multiple species of operators,
as in [71]. Indeed, a generic CFT, as opposed to a bottom-up generalized free eld
theory, always has an in nite number of single-trace operators. If there are additional
elds
in the bulk with four-point couplings ~4 2 2, then the corresponding bubble
diagrams are also easy to compute, given the tree-level hOOO O i; see [26]. When
16Note that for operators of
2 Z, there is potential mixing between [OO] and [T T ], at least in some
channel where global symmetry-singlet [OO] operators contribute.
there are also three-point
vertices, the situation is more complicated. The case
when some [O O ] operator is degenerate with a [OO] operator with the same
quantum numbers is discussed in appendix A.1, and requires generalizing the bootstrap
analysis to di erent external operators. It should be straightforward to understand
the form of M1 loop when external dimensions are unequal. The basic structure will
be identical to (3.23): M1 loop will have poles at
= 2
+ 2n in every channel, with
residues xed by rst-order data.
An extension to higher loops would also be pro table. On the CFT side, one will
generally have to contend with triple- and higher-trace operators. By the arguments
of section 3, one sees that at O(1=N 2(L+1)) | dual to L-loop order in AdS | ML loop
has poles of degree
information from
L. In general theories, going to O(1=N 6) requires additional
ve-point functions. However, in theories that have a O $
O
symmetry, these ve-point functions vanish (and correspondingly no triple-trace
operators appear in the O
O OPE). So in these theories it may be possible to compute
the four-point functions and anomalous dimensions also at two-loop order, with no
new conceptual wrinkles.
What we really seek, however, are the AdS loop-level Feynman rules for Mellin
amplitudes, thus giving an algorithm for any L-loop calculation.
In general the 1=N expansion is only asymptotic, and there are non-perturbative
e ects scaling as (say) e N that must be understood before even attempting to
continue the large N results to nite values of N . Can we use our methods also for such
non-perturbative contributions?
As we discussed, the crossing analysis simpli es considerably when the dimension
is an integer. It would be interesting to analyze the case of non-integer
and to
obtain explicit results for this case as well.
For theories with three-point vertices, we found (see (2.30)) that the four-point
function at O(1=N 4) has a contribution proportional to u log2(u) log2(v), with a
crossing-symmetric coe cient function h(u; v). What are the form and content of
this function?
Our analysis in this paper did not assume any additional symmetries. It should
be simple to take into account additional global symmetries. Incorporating
supersymmetry should also be straightforward, at least in principle, with superconformal
blocks replacing the conformal blocks.
An especially interesting example, as always, is the d = 4, N = 4 SYM theory. In the
2
Y M = gY M N ! 1 limit, the bulk theory only contains the elds dual to protected
single-trace operators.
The solution to crossing in this limit at order 1=N 2 was
performed in [39]. Its generalization to order 1=N 4 involves all the issues mentioned
earlier: in particular, there is an in nite number of single-trace operators, and they
all have integer dimensions so that there can be complicated mixings between the
HJEP07(21)36
various double-trace operators. Luckily, the four-point functions of all these protected
single-trace operators were recently computed in [66], and this information should
be su cient to work out the mixing matrix, and thus to compute the correlation
functions of protected operators in this theory at order 1=N 4. It would be interesting
to perform this analysis.
In fact, the analysis should be simpler than it may appear. We now make a potentially
powerful observation.
Consider the Mellin amplitude for the four-point function
hO200 O200 O200 O200 i. At large Y M , all single-trace operators in the O200
O200 OPE
have even twist: these are the operators Ok with even k
BPS operators in the [0; k; 0] representation of SU(4). (O200
2, which are the 1/2
O2.) Due to
nonrenormalization of the Ok dimensions and SU(4) selection rules, the only operators
appearing in M1 loop will be the double-trace operators [OkOk]n;`, with k
2; so all
poles in M1 loop sit at even twist
= 4 + 2n. Now, for every value of n, there are
double-trace operators [O200 O200 ]n;`. If we compute M1O2l0o0oOp200 for this correlator |
that is, the piece of M1 loop
xed by requiring a match to the contributions of the
[O200 O200 ]n;` operators, as done in section 3 | the residues at all twists
= 4 + 2n
are
xed by [O200 O200 ]n;` tree-level data. Therefore, up to regular terms, this xes
the full one-loop amplitude! That is, up to regular terms,
M1 loop = M1[O2lo00oOp200 ] :
(7.1)
While the operator mixings mentioned above still plague the calculation, (7.1) says
that the full one-loop amplitude | which involves an in nite set of diagrams involving
virtual Ok loops | is determined just by the anomalous dimensions
[O200 O200 ]n;` operators. This is a great simpli cation, apparently due to the e ect of
n(1;`) for the
maximal supersymmetry on the spectrum.
In general, for a four-point function of some operator O, this coincidence of poles
occurs whenever
O 2 Z and the spectrum of twists in the O
Besides N = 4 SYM, this also occurs when O is the bottom component of the stress
tensor multiplet of the d = 6, N = (2; 0) theory of M5-branes. We can also analyze
many other interesting supersymmetric conformal eld theories, such as the d = 3,
N = 8 theory of M2-branes (which does not have the same simpli cation described
above). In both of these cases, there is again a gap to the non-protected operators, but
here it scales as a power of N that does not involve an extra independent parameter.
Thus one cannot separate the loop expansion and the derivative expansion in the
bulk.18 In any case, the loop diagrams in the dual AdS bulk can still be computed
O OPE is even.17
by the methods described in this paper.
17This phenomenon has a tree-level version: in a tree-level exchange of twist between external operators
of dimension
, the amplitude has only a
nite number of poles when
2
2 2Z. This happens because
the single-trace and double-trace poles collide, and would thus produce a triple pole, violating the 1=N
expansion, unless these single-trace poles drop out of the amplitude. This was also recently noted in [66].
18At some speci c low orders in 1=N , it is possible to separate the di erent contributions to the correlation
functions: in particular, the leading 1=N correction is due not to a loop, but to a higher-derivative correction
to the action that descends from anomalies in d = 11 supergravity (e.g. [72]).
7.2
When we have a standard eld theory in AdS space (as opposed to a gravitational
one), it has not just correlation functions with sources at the boundary as we discussed
in this paper, but also correlation functions of operators at arbitrary bulk points. Are
these determined in terms of the correlation functions with boundary sources? Can
we say anything about them by our methods?
In our discussion of the N = 4 SYM theory we integrated out the stringy states, but
we can repeat the same story when including non-protected string states. For nite
Y M there are additional operators contributing with dimensions at least of order
1=4 . These operators can be integrated out in an expansion in 1= 1Y=M4 , whose form
Y M
at order 1=N 2 was discussed in [39]. Using this information it should be possible to
work out also the order 1=N 4 correlators in a systematic expansion in 1= 1Y=M4 . Can
we use the large N expansion of the crossing equation to learn anything about the
non-protected states?
For bulk theories which are string theory backgrounds, the 1=N expansion (the loop
expansion in the bulk) coincides with the genus expansion of the worldsheet theory.
The correlators we discuss arise as integrated correlation functions in this worldsheet
theory. What does our analysis teach us about these worldsheet theories? Can we
relate the crossing equations in the CFT and in the worldsheet theory?
We close with some words on the relation of the large N bootstrap to at space
physics. An alternative way to approach the AdS amplitudes problem might have
been to start from known facts about S-matrices, and
nd analogs or extensions to
AdS. We took a di erent tack, but it would be very interesting to turn to these
questions using our results. In [26], the emergence of the optical theorem in the at
space limit was studied, but one would also like to know whether there is a direct
analog at nite AdS curvature.
Similarly, in our one-loop crossing computations we found speci c harmonic
polylogarithms appearing. As we noted, this suggests an intriguing underlying structure
akin to
at space amplitudes. On the other hand, the one-loop Mellin amplitudes
themselves were given by the more familiar generalized hypergeometric functions and,
in the case of (5.41), a digamma function. What class of functions forms a basis for
the multi-loop solution of the crossing equations, and for the AdS Mellin amplitudes
themselves? Which diagrams form a basis for all others at a given loop order? The
answers would presumably be closely related to the possible existence of AdS analogs
of generalized unitarity, on-shell methods and the like. It would be fascinating to try
to understand the big picture here.
Finally, we note that Mellin amplitudes admit at space limits [15]. If one can develop
the solution to crossing to successively higher orders in 1=N , taking that limit would
shed light on at space higher-loop amplitudes. A speci c, and di cult, longer-term
challenge in the supergravity community is to determine the critical dimension above
HJEP07(21)36
which the four-point, ve-loop amplitude in maximal supergravity diverges. This
has resisted years of direct attack using advanced methods [73{75]. It would be
fascinating if, eventually, the ve-loop crossing equations, applied to the holographic
dual of gauged maximal supergravity, could be employed in this endeavor.
Acknowledgments
We wish to thank Nima Afkhami-Jeddi, Michael B. Green, Vasco Goncalves, Tom Hartman,
Zohar Komargodski, David Simmons-Du n, Ellis Ye Yuan and Sasha Zhiboedov for helpful
discussions. The work of OA was supported in part by the I-CORE program of the Planning
and Budgeting Committee and the Israel Science Foundation (grant number 1937/12),
by an Israel Science Foundation center for excellence grant, by the Minerva foundation
with funding from the Federal German Ministry for Education and Research, by a Henri
Gutwirth award from the Henri Gutwirth Fund for the Promotion of Research, and by the
ISF within the ISF-UGC joint research program framework (grant no. 1200/14). OA is
the Samuel Sebba Professorial Chair of Pure and Applied Physics. The work of LFA was
supported by ERC STG grant 306260. LFA is a Wolfson Royal Society Research Merit
Award holder. AB acknowledges the University of Oxford for hospitality where part of
this work has been done. AB is partially supported by Templeton Award 52476 of A.
Strominger and by Simons Investigator Award from the Simons Foundation of X. Yin. EP
is supported by the Department of Energy under Grant No. DE-FG02-91ER40671.
A
Operator content of the one-loop crossing equations
In this appendix we discuss the operators that can appear in the OPE of two identical
single-trace primary operators O and O of dimension
expansion they contribute to the crossing equation. The upshot is that at order 1=N 4, we
do not have to consider any operators with more than two traces appearing in the OPE.
, and at which order in a large N
The notation is that [O1O2 : : : Om] is an m-trace primary operator corresponding
to an m-particle state in the bulk (and appearing at N
Om(xm); for the precise de nition at m = 2, see appendices of [15, 27]).
All operators will be normalized such that their two-point function is one. We will choose
a basis in which there is no mixing between operators with a di erent number of traces (we
will discuss mixings of di erent double-trace operators below). This means, for instance,
that [OiOj ] is not exactly the operator appearing in the OPE of Oi and Oj , but may di er
from it at order 1=N ; these di erences will not be important in the order we work in.
On general grounds, connected n-point functions of single-trace operators scale as
1=N n 2 in the large N limit. Naively this implies that the OPE coe cient of a k-trace
operator, proportional to hOO[O1
Ok]i scales as 1=N k. In general this expectation can
fail only if there is an extra disconnected contribution to this correlation function. However
in our case, since we chose the single-trace operators to be orthogonal to operators with
more traces, such a disconnected correlation function can only appear for the operators
[OO], which have OPE coe cients of order one. Thus the OPE coe cient of operators
1 in the OPE of
with three or more traces is suppressed at least by 1=N 3, so they will not contribute to the
crossing equation at order 1=N 4.
The only operators contributing at order 1=N 4 are then:
Single-trace operators O1, with some even spin ` (a special case is the
energymomentum tensor): the OPE coe cient cOOO1 is generically of order 1=N , so they
contribute to crossing already at order 1=N 2. At order 1=N 4 we will see corrections
to these contributions due to 1=N 2 corrections to the dimensions of O and O1, and
to cOOO1 . These cannot be determined by crossing since they are the basic inputs
| in the bulk these are masses and three-point vertices that need to be determined
by some renormalization condition at all orders in 1=N . Thus from the point of view
of the crossing equation we need to take these as given. If we use renormalization
conditions that are independent of N , and in particular for protected operators in
superconformal eld theories, single-trace operators will appear in the crossing
equation only at order 1=N 2; otherwise their contributions at higher orders are simply
related to the leading order contribution and to the corrections to the dimensions
and single-trace OPE coe cients.
coe cients a(n0;`), and with dimensions 2
Double-trace operators [OO]n;`: these appear already at order 1 with squared OPE
+ 2n + `. As we discuss extensively, at
higher orders in 1=N they give contributions related to the corrections to the OPE
coe cients and dimensions of these double-trace operators.
Other double-trace operators [O1O2]n;l: the OPE coe cient cOO[O1O2]n;l is of order
1=N 2. Thus, generally these operators appear in the crossing equation at order 1=N 4,
with a contribution depending on the leading order dimension
1 +
2, and on the
leading order cOO[O1O2]n;l . The latter depends on four-point couplings in the bulk
which are arbitrary, so from the point of view of the four-point function hOOOOi
they will give us parameters that we cannot determine. However, because these
contributions depend only on the leading order dimensions, they generically do not
come with any logs in the direct channel, so they will not a ect the universal terms
that we discuss in this paper; they give rise to independent poles in Mellin space. This
is not true when these operators mix with the [OO] operators, as we discuss below.
At order 1=N 6 the analysis will change, and in particular triple-trace operators will also
start appearing, depending on (undetermined from crossing) ve-point vertices in the bulk.
A.1
Degeneracies
One important issue that was ignored in the analysis above is mixing between di erent
double-trace operators when they are degenerate; this often happens in interesting
examples, and a mixing of [OO] with other double-trace operators signi cantly modi es
the analysis.
As a typical example, consider a 2 21 eld theory on AdS, where
and 1 are scalars
with the same mass, and where O is dual to
and O1 to 1. In this theory, the OPE of
HJEP07(21)36
The resolution is that the two double-trace operators mix: there is a bulk tree-level
O and O contains [OO] starting at order 1 from the disconnected diagram in AdS, and
[O1O1] starting at order 1=N 2 from an X-shaped diagram, and no single-trace operators.
There are no tree-level diagrams contributing to hOOOOi and to hO1O1O1O1i, so naively
the analysis at order 1=N 2 implies that [OO] and [O1O1] have no anomalous dimensions at
this order. We then expect to have no logarithmic terms in the direct-channel four-point
function at order 1=N 2, and no double-logs at order 1=N 4 (i.e. no poles in M1 loop). But
on the other hand, the one-loop diagram contributing to hOOOOi is clearly the same as
in the 4 theory, which does have such double-logs/poles since the latter theory does have
a non-trivial tree-level diagram.
operator (1) =
B
diagram giving a non-zero h[OO][O1O1]i
diagonal two-point functions is OA
turns out that the rst operator has an anomalous dimension
1=N 2. The correct basis of operators with
[OO] + [O1O1] and OB(1) = C=N 2 and the second
[OO] [O1O1], and it
A
C=N 2, for some constant C. This reproduces the two-point functions at
order 1=N 2. It also explains why we get double-logs at order 1=N 4 (poles in M1 loop) in
hOOOOi, since these are proportional to ( A(1))2 + ( B(1))2 (both OA and OB appear in the
O
O OPE), which is non-zero.
The lesson is that in general, we have to be careful of double-trace mixings; all operators
that mix with [OO]n;` appear in the crossing equation already at order 1=N 2, and will lead
to double-logs at order 1=N 4. The coe cients of these double-logs cannot be computed
without knowing the precise mixing matrix: one has to know all correlators h[OO][O1O2]i
at order 1=N 2, which can be extracted from tree-level hOOO1O2i four-point functions,
before one can use the crossing equation at order 1=N 4. Note that mixings of this type
occur in the N = 4 SYM theory, complicating its analysis.
B
Explicit expansions
In this appendix we display explicit results for the large spin expansion for 0(2;`) in several
examples. Recall that the total result is the sum over contributions from each conformal
block in the dual channel. In a case in which the solution at order 1=N 2 has support only
for spin zero we obtain
(2)
0;` (n;s)
(0)
cn;s
(2)
J 2 +2n ^0;` (n;s)
:
(` + 2)2(`(`(6`(` + 7) + 115) + 148) + 77) + 3J 6 3`2 + 9` + 8
(2)(` + 1) ;
(` + 1)(` + 2)3(`(`(3`(`(10`(` + 10) + 443) + 1111) + 4975) + 4187) + 1558)
+ J
5 8 5`4 + 30`3 + 79`2 + 102` + 54
(2)(` + 1) ;
^0;` (n;0)
For
^0;` (0;0)
^0;` (1;0)
^0;` (2;0)
3
2
5
36
6
0(2;`) =
8
1 X a(n0;0)
(1) 2 (2)
0;` (n;0)
has the structure explained in section 4.
= 2 in d = 4 we obtain for the rst few cases
2(` + 2)(`(` + 4) + 5)
` + 1
+ 2J 4 (2)(` + 1) ;
(B.1)
(B.2)
where we have introduced J 2 = (` + 1)(` + 2). As explained in the body of the paper, with
these ingredients it is possible to obtain
zero, we get
0(2;`) also for nite values of the spin `. For spin
36(1 + n)4(5 + 2n(3 + n))
(3 + 4n(2 + n))
72(1 + n)8
3 + 4n(2 + n)
(2)(n + 1)
2
:
(B.3)
= 2 in d = 2, equivalently one can nd the form of ^0(2;`)j(n;0) and compute 0(2;`).
As already discussed, this sum is divergent.
For spin zero in this case we obtain
0(2;0) = X
9(19+n(25+2n(6+n)))
2(1+n)
better organised in powers of J 2. For instance, for the interaction
4 with
For each model we can obtain the expansion of 0(2;`) around large `. The expansion is
= 2 in d = 4
+ 9(1+n)2(2+n)2
2 =
2
:
(B.4)
HJEP07(21)36
18 1
5 J 2 +
96 1
7 J 4 +
360 1
7 J 6 +
74304 1
724320 1
1001 J 10 +
2
;
(B.5)
for the interaction 4 with
= 2 in d = 2 we obtain
1 +
4 1
5 J 2 +
4 1
7 J 4 +
16 1
35 J 6 +
16 1
55 J 8 +
1856 1
5005 J 10 +
while for the mixed interaction 33! 3 + 44! 4 with
= 2 in d = 4 we obtain
96 1
72 1
576 1
The expansions above are asymptotic. In the body of the paper we have shown how
to resum the expansions and compute them for nite values of the spin. It is interesting
to compare the asymptotic series above with the correct results for di erent values of the
spin. For instance, for ` = 2 we have J 2 = 12. Including the rst six terms shown above
for 4 and 3 + 4 in d = 4 we would obtain
2
;
(B.6)
(B.7)
(B.8)
(B.9)
(B.10)
(B.11)
(2)
0;2
(2)
0;2
0:11977 2;
0:591144 32 4;
for 4
for 3 + 4
to be compared with the exact values
(174 2
;
(39 2
350) 32 4
0:591146 32 4;
for 3 + 4
:
We see that the values we obtain from the asymptotic series are remarkably close to the
correct values, even for spin two! In the case of convergent answers, even the approximation
for spin zero is very good.
When we compute bulk loop diagrams we expect to get UV divergences. Since these arise
at short distances, they should take a similar form in AdS as in
at space, and at any
loop order we should be able to cancel them by local counter-terms in AdS. In general
the bulk theories we discuss are e ective theories which are non-renormalizable, so they
require a cuto , and at higher orders in perturbation theory we will need to add more and
more counter-terms, but in this paper we just discuss the one-loop order. As argued in the
Introduction, in our bootstrap computation related to a divergent bulk diagram we expect
to nd a divergence in
n(2;`), and we expect that when we regularize it (for instance by
putting some cuto on the sums), the divergence is precisely proportional to
from some local bulk terms, so that it can be removed by putting in appropriate cuto
n(1;`) coming
dependent bulk terms.
Recall that on general grounds we expect any local bulk term that is allowed by the
symmetries to appear with an arbitrary coe cient, both from the bulk point of view, and
from the bootstrap point of view, since any such term gives a solution to the crossing
equations. Thus at any loop order any solution that we nd for the four-point function,
both from the eld theory and bootstrap points of view, is just up to bulk terms. This
means that we should take both the dimensions and the three-point functions of single-trace
operators, at all orders in 1=N , to be inputs to the computation, that we cannot determine
just from the crossing equations in a 1=N expansion. In addition we have a freedom to
choose any local four-point terms, namely to shift the solution by any of the \homogeneous"
solutions to the crossing equations that correspond to
nite-order polynomials in Mellin
space (we called them freg in section 3.3.1). We expect to need this freedom in order to
cancel divergences. We cannot x it just from crossing.
Consider rst the
4 theory in AdS5. The coupling constant
here has dimensions
of length, and one can de ne a dimensionless coupling =RAdS, that in our 1=N expansion
is proportional to 1=N 2.
In
at space the four-particle tree-level scattering amplitude goes like ; when we
translate it into some dimensionless quantity this will go at high energies as
E where E
is a typical energy. In AdS the role of the energy is played by n, so we expect to nd for
the tree-level four-point amplitude a result going as
(1)
n;` /
n ' N 2
(C.1)
at large n, which is indeed what we nd (5.1). (In this case the answer happens to vanish
for l > 0.) Note that large n here means n
1 and n
, so that the energy is larger
than the mass and the scale of the AdS radius.
At one-loop in
at space we have a linear divergence, and the amplitude with a nite
cuto
goes at high energies as 2
+ E +
). Note that we do not get a logarithmic
divergence; indeed such a divergence would multiply E but there is no local counter-term
that could cancel this (higher-derivative couplings in the bulk give higher powers of E).
Noting that the divergence is just a constant, it can be canceled by shifting
by a term
proportional to
2 . Translating to AdS as above, we expect to
nd for the one-loop,
four-point function at large n
n;` /
2(n2 + ~ n) '
n2 + ~ n
N 4
where ~ is some cuto
large n behavior of n(2;`) in 4
that we use to obtain a nite result. This is a prediction for the
. We expect from the locality of the divergence that we could
obtain a nite result by shifting n(2;`) by a term proportional to ~ n(1;`) of the 4 theory. Note
in particular that this means that only ` = 0 terms should diverge, and this is indeed what
we nd in section 5. Note also that as far as the crossing equations in the 1=N expansion
are concerned, there is no obvious way to x the nite local 4 bulk term remaining after
this subtraction.
In general we get precise predictions for which divergences we should get in our
computation. It should always be possible to cancel divergences in
4- counter-terms by adding terms proportional to the n;` 's that are associated with the
counter-terms we need in the bulk. In Mellin space these divergences should always be
a polynomial, of a
nite degree related to the loop order. Above one-loop, divergences
(1)
related to counter-terms with more 's can also appear.
The analysis of the N = 4 SYM theory, and the related supergravity on AdS5, is
analogous. The only di erence is that we have to be careful if we regularize our computation,
that the regularization preserves supersymmetry, otherwise we will get divergences that are
related to bulk counter-terms that are di erent from the supersymmetric local terms in the
bulk. Using a supersymmetric regularization the divergences should all be proportional to
the tree-level contributions analyzed in [39]. We leave a detailed discussion of this case to
n(k;`) that are related to
Finally, if we consider the 3 theory in AdS5 (d = 4), the theory is super-renormalizable
so there should be no divergences in the four-point functions that cannot be swallowed
into the masses and three-point couplings in the bulk. In this case dimensional analysis
implies that all n(k;`) should not grow at large n, and that we would not encounter any UV
divergences in their computation. For d > 5 the bulk theory is non-renormalizable, so we
expect the large n behavior of the tree-level terms to go as nd 5, and one-loop terms to go
as the square of this.
D
3 OPE data
In this appendix we derive the tree-level anomalous dimensions of double trace operators
due to a fully symmetric exchange of a scalar operator of
= 2, i.e. for a
= 2 scalar
with a 33! 3 coupling in AdS. This result was quoted in (5.27) for ` = 0.
Such an exchange has been considered in [11, 15] and it can be reduced to
G(u; v) = (d)u2D1212(u; v) ;
(D.1)
D i (u; v) =
2
i ( i
)
i i
2) x123 1 x224 2
where (d) is a constant which depends on the number of space time dimensions, in
particular (4) = 8 23 = 2C2
and (2) = 2 23 = C2
OOO. The functions D(u; v) are de ned as
1 2 3
2 2 4
2
x13
x214x34
2
D i (xi) ; (D.2)
2
x14
x213x34
2
HJEP07(21)36
G(u; v) = G(u; v) + G
= (d)u2 D1212(u; v) + D2211(u; v) + D1221(u; v) ;
where in the last line the symmetry properties of D(u; v) have been used. To compute the
anomalous dimension it is enough to focus on the terms proportional to log u in (D.8) and
perform the conformal partial wave expansion
G(u; v)jlog u =
2
1 X an;` n;`
(0) (1); 3 un+2g4+2n+`;`(u; v) :
Notice that the small u expansion of G(u; v) starts at order u, but this contribution does
not contain any log u. This is consistent with the expectations, since the OPE contains
the scalar operator of exact dimension two and all its descendants. It is straightforward to
extract the anomalous dimension from (D.9), both for d = 2 and d = 4:
= D 3 2 1 4 (v; u)
= u
2 D 4 2 3 1
u 1
v v
+ u2G
1 v
u u
u 1
v v
1 v
u u
:
(D.4)
(D.6)
(D.8)
(D.10)
HJEP07(21)36
where
The symmetry properties of D(u; v) are
D i (xi) =
2
i i
i ( i
2) Z 1 Y dtiti i 1e 21 Pi;j titjxi2j :
D 1 2 3 4 (u; v) = v
2 D 1 2 4 3
We would like to study the fully symmetrized amplitude which corresponds to
d = 2 :
d = 4 :
(
2(5+4n)
(1+n)(2+n)(3+2n) 3
4
(`+1+n)(`+2+n) 3
2(2 7(1+n)2)
(1+n)(3+4n(2+n)) 3
(`+1)(`+2+2n) 3
2 ; ` = 0
2 ; ` 6= 0
2 ; ` = 0
2 ; ` 6= 0
Open Access.
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any medium, provided the original author(s) and source are credited.
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